Localizing virtual cycles by cosections

Low degree GW invariants of surfaces II

“…Also, when X is proper, the localized GW-invariants coincide with the ordinary GW-invariants of X [16] (see also [8,Lemma 6.5]). Example 2.5.…”

“…Because the construction of cosection localized virtual cycles has shown larger potential in applications in a wider range of problems, we have grouped the basic construction of cosection localized virtual cycles, the algebraic construction of localized Gysin map, and the application to GW-invariants of surfaces with holomorphic two-forms in the preprint [8].…”

“…[8] The locus Z(σ θ ) (i.e., the non-surjective locus of σ θ ) consists of all θ-null stable morphisms in M χ,n (X, β) • .…”

“…We will prove a deformation invariance of localized GW-invariants of surfaces, prove a degeneration formula of localized GW-invariants of spin surfaces, and in the end prove the formulas of low degree GW-invariants of surfaces with positive p g conjectured by Maulik-Pandharipande. In [7], in our attempt to understand Lee-Parker's work [16] on GW-invariants of Kähler surfaces with positive p g , the authors constructed the cosection localized virtual class for a DM stack with a perfect obstruction theory and a cosection of its obstruction sheaf; the algebro-geometric construction of such cycles was completed after constructing the algebraic localized Gysin map [8]. Applied to the moduli of stable morphisms to surfaces, this reproduces Lee-Parker's localization of GW-invariants of surfaces with positive p g .…”

“…For c ∈ T a closed point and a class β = 0 ∈ H 2 (X c , Z), we let Θ c ∈ Γ(Ω 2 Xc ) be the pull-back of Θ to X c ; we form the moduli spaces of stable morphisms M χ,n (X , β) • and M χ,n (X c , β) • . Using the two-form Θ, and applying [8,Sect. 6], we obtain cosections σ Θ and σ Θc of the respective obstruction sheaves of M χ,n (X , β) • and M χ,n (X c , β) • , and then their respective cosection localized virtual classes.…”